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URL:https://uwaterloo.ca/combinatorics-and-optimization/events/algebraic-gr
 aph-theory-shivaramakrishna-pragada
SUMMARY:Algebraic Graph Theory-Shivaramakrishna Pragada
CLASS:PUBLIC
DESCRIPTION:TITLE: Structure of Eigenvectors of Graphs \n\nSPEAKER:\n Shi
 varamakrishna Pragada\n\nAFFILIATION: \n\nSimon Fraser University\n\nLOCAT
 ION:\n Please contact Sabrina Lato for Zoom link.\n\nABSTRACT: Let G be
  a graph on n vertices with characteristic\npolynomial φ_G(λ). A graph i
 s said to be irreducible if the\ncharacteristic polynomial of its\n\nadjac
 ency matrix is irreducible. For every irreducible graph G\, we\nshow that 
 each eigenvector of its adjacency matrix has pairwise\ndistinct\, non-zero
  entries. \nMore generally\, consider a graph G whose characteristic polyn
 omial\nfactors over Q as φ_G(λ) = p_1(λ)· · · p_k(λ)\, where the\np
 olynomials p_i(λ) are distinct irreducible factors. For any\neigenvalue 
 θ with minimal polynomial p_j (λ)\, we prove a structure\ntheorem of eig
 enspaces corresponding each polynomial p_j (λ). We\nderive a lower bound 
 on the number of distinct entries that must\nappear in every eigenvector c
 orresponding to θ. \nIt is conjectured that almost all graphs have irredu
 cible\ncharacteristic polynomials\, this has recently been confirmed under
  the\nassumption of the Extended Riemann Hypothesis. We pose new structura
 l\nquestions about irreducible graphs and present preliminary progress\nto
 ward understanding their eigenvectors and spectral properties.
DTSTAMP:20260402T170603Z
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